It follows from the definitions that the formations of \(P_\mathfrak {G}(\lambda ),U_\mathfrak {G}(\lambda )\)  and \(Z_\mathfrak {G}(\lambda )\)  commute with any base extension on \(k\) . In particular, for every \(s\in \operatorname{Spec}k\)  we have \(P_\mathfrak {G}(\lambda )_{\bar{\kappa }(s)}=P_{\mathfrak {G}_{\bar{\kappa }(s)}}(\lambda _{\bar{\kappa }(s)})\) , which is a parabolic subgroup of \(\mathfrak {G}_{\bar{\kappa }(s)}\)  by [1]}. Hence, \(P_\mathfrak {G}(\lambda )\)  is a parabolic \(k\) -group.
